Spatial statistics and ESDA
GEOG 40323
February 27, 2018
Statistics
- Definition: the study of the collection, organization, analysis, interpretation, and presentation of data.
- In turn, an understanding of statistics is fundamental to the GIS practitioner
Measures of central tendency
Mean
The mean of a sample (\(\overline{x}\)) is calculated as follows:
\[\overline{x} = \dfrac{x_1 + x_2 + ... + x_n}{n}\]
where \(n\) is the number of elements in the sample.
Mode
- Most frequent value in a sample
Variance
- A measure of the spread of a sample. The variance is computed as:
\[{\sigma}^2 = \dfrac{\sum\limits_{i=1}^{n}(x_i - \overline{x})^2}{n}\]
or, in simpler terms, the average of the squared deviations of the values of a sample from its mean.
Standard deviation
- Computed as the square root of the variance; denoted by \(\sigma\).
- Offers a standardized way to discuss the spread of a distribution. For example, in a normally distributed sample (more to come on this):
- About 67 percent of the values will be within one standard deviation of the mean
- About 95 percent of the values will be within two standard deviations of the mean
- About 99 percent of the values will be within three standard deviations of the mean
Z-score
- Often, variables in your analysis will have vastly different measurement scales. Variables can thus be standardized with the computation of z-scores.
- A z-score is computed as follows:
\[Z = \dfrac{x - \overline{x}}{\sigma}\]
- In turn, a z-score reflects how many standard deviations an observation is away from the mean.
Probability
- Probability: the likelihood of the occurrence of an event
- Statistical significance: the probability that an observation/effect is not the result of random chance
The normal distribution
Implications of the distribution of your data
Correlation
- Generally speaking, correlation refers to the extent to which two variables covary.
- The most popular measure of correlation is Pearson’s product-moment correlation coefficient (or Pearson’s \(r\)), which is appropriate for two continuous variables.
- \(r\) ranges from -1 to 1, where -1 reflects an inverse relationship between two variables, and 1 reflects a positive relationship. If \(r\) = 0, no apparent relationship exists.
Regression
- Regression analysis is used to study the relationship between a response variable \(Y\) and a series of explanatory variables \(X_1 ... X_n\).
- The general formula of a linear regression function is:
\[Y = a + b_{1}X_{1} + ... + b_{n}X_{n} + e\]
- Our job is to estimate the unknown intercept \(a\) and the parameters \(b_1 ... b_n\). \(e\) refers to the error, or residuals, in our equation.
Spatial statistics
- Statistical methods make a variety of assumptions about your data; for example, regression assumes that your residuals (errors) are independent and identically distributed (\(i.i.d\)).
- However, we remember Tobler’s First Law of Geography:
- Given this property of spatial data, we need special methods to account for spatial dependence.
Exploratory spatial data analysis (ESDA)
- Data analysis techniques used to investigate spatial patterns in a dataset
- Used to discover unseen features of datasets; often involves visualization
Standard deviation ellipse
- Reveals the directional distribution of points; analogous to a standard deviation
Nearest neighbor
- Compares the average distance from a feature to its neighbor to an expected average under conditions of spatial randomness
Ripley’s K function
- Checks for clustering/dispersion at multiple scales
Spatial autocorrelation
- Autocorrelation: the correlation of a variable with itself
- Temporal autocorrelation: similar values in similar time periods
- Spatial autocorrelation: when features tend to have similar values to their neighbors (Tobler’s first law of geography!)
Spatial weights matrix
- A spatial weights matrix is used to conceptualize spatial relationships between features
- Types: distance-decay, contiguity, nearest-neighbor
Global measures of autocorrelation
- Statistics designed to detect the presence of spatial autocorrelation in a dataset
- Commonly used: Moran’s I, Geary’s C
Moran’s I
- Most commonly-used global autocorrelation statistic
- Values range from -1 to 1; negative values indicate uniformity, positive values indicate clustering
Local measures of spatial autocorrelation
- Local statistic: statistic in a spatial dataset that varies from place to place
- Whereas global measures of spatial autocorrelation can describe general characteristics of a dataset, local measures can illustrate where clustering is found in your data
Getis-Ord \(G_i\)
- For a location \(i\), the local statistic \(G_i\) is computed as:
\[G_{i}(d) = \dfrac{\sum\limits_{j}w_{ij}(d)x_j}{\sum\limits_{j=1}^{n}x_j} \text{ for all }i \neq j\]
- In simpler terms, \(G_i\) reflects the relationship between the values of a neighborhood and the overall values of a study area. High values reflect a clustering of high values; low values reflect a clustering of low values.
- The \(G_i^{*}\) statistic is similar, but includes the value for location \(i\) in the neighborhood calculation.
Local indicators of spatial association (LISA)
- Local form of the Moran’s I; used to identify clusters and outliers in a dataset
Spatial regression
- Spatial dependence in your data violates some basic assumptions of the regression model
- Alternative: spatial regression
- Spatial lag model: The average neighborhood value of the response variable modeled as an explanatory variable
- Spatial error model: Used when residuals (errors) exhibit spatial autocorrelation; accounts for spatial autocorrelation in the error term
Spatial statistics in ArcGIS
Other options for doing spatial analysis
